Algorithm for sorting bit sequences in linear complexity

ABSTRACT

A computer program product and associated algorithm for sorting S sequences of binary bits. The S sequences may be integers, floating point numbers, or character strings. The algorithm is executed by a processor of a computer system. Each sequence includes contiguous fields of bits. The algorithm executes program code at nodes of a linked execution structure in a sequential order with respect to the nodes. The algorithm executes a masking of the contiguous fields of the S sequences in accordance with a mask whose content is keyed to the field being masked. The sequential order of execution of the nodes is a function of an ordering of masking results of the masking. Each sequence, or a pointer to each sequence, is outputted to an array in the memory device whenever the masking places the sequence in a leaf node of the nodal linked execution structure.

This application is a continuation application claiming priority to Ser.No. 10/696,404, filed Oct. 28, 2003.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention generally relates to an algorithm for sorting bitsequences, and in particular to an algorithm for sorting bit sequencesin linear complexity.

2. Related Art

In the current state of the art with respect to sorting words (i.e.,integers, strings, etc.), the fastest known algorithms have an executionspeed proportional to N_(W) log N_(W) (i.e., of order N_(W) log N_(W)),wherein N_(W) denotes the number of words to be sorted. The well-knownQuicksort algorithm is an in-place sort algorithm (i.e., the sorteditems occupy the same storage as the original items) that uses a divideand conquer methodology. To solve a problem by divide and conquer on anoriginal instance of a given size, the original instance is divided intotwo or more smaller instances; each of these smaller instances isrecursively solved (i.e., similarly divided), and the resultantsolutions are combined to produce a solution for the original instance.To implement divide and conquer, Quicksort picks an element from thearray (the pivot), partitions the remaining elements into those greaterthan and less than this pivot, and recursively sorts the partitions. Theexecution speed of Quicksort is a function of the sort ordering that ispresent in the array of words to be sorted. For a totally randomdistribution of words to be sorted, Quicksort's execution speed isproportional to N_(W) log N_(W). In some cases in which the words to besorted deviate from perfect randomness, the execution speed maydeteriorates relative to N_(W) log N_(W) and is proportional to (N_(W))²in the worst case.

Given, the enormous execution time devoted to sorting a large number ofintegers, strings, etc. for extensively used applications such asspreadsheets, database applications, etc., there is a need for a sortalgorithm having an execution speed of order less than N_(W) log N_(W).

SUMMARY OF THE INVENTION

The present invention provides a method, computer program product, andassociated algorithm for sorting S sequences of binary bits in ascendingor descending order of a value associated with each sequence and in atime period denoted as a sorting execution time, said S sequences beingstored in a memory device of the computer system prior to said sorting,S being at least 2, each sequence of the S sequences comprisingcontiguous fields of bits, said sorting comprising executing programcode at nodes of a linked execution structure, said executing programcode being performed in a sequential order with respect to said nodes,said executing including:

masking the contiguous fields of the S sequences in accordance with amask whose content is keyed to the field being masked, said sequentialorder being a function of an ordering of masking results of saidmasking; and

outputting each sequence of the S sequences or a pointer thereto to anoutput array of the memory device whenever said masking places said eachsequence in a leaf node of the linked execution structure.

The present invention provides a method, computer program product, andassociated algorithm for sorting S sequences of binary bits in ascendingor descending order of a value associated with each sequence and in atime period denoted as a sorting execution time, said S sequences beingstored in a memory device of the computer system prior to said sorting,S being at least 2, each sequence of the S sequences comprising Kcontiguous fields denoted left to right as F₁, F₂, . . . , F_(K) withcorresponding field widths of W₁, W₂, . . . , W_(K), said sortingcomprising the steps of:

designating S memory areas of the memory device as A₁, A₂, . . . ,A_(S);

setting an output index P=0 and a field index Q=0;

providing a node E having S elements stored therein, said S elementsconsisting of the S sequences or S pointers respectively pointing to theS sequences; and

executing program code, including determining a truth or falsity of anassertion that the elements in node E collectively include or point tono more than one unique sequence U of the S sequences, and if saidassertion is determined to be false:

-   -   then generating C child nodes from node E, each child node        including all elements in node E having a unique value of field        F_(Q+1), said child nodes denoted as E₀, E₁, . . . , E_(C−1)        having associated field F_(Q+1) values of V₀, V₁, . . . ,        V_(C−1), said child nodes E₀, E₁, . . . , E_(C−1) being        sequenced such that V₀<V₁< . . . <V_(C−1), said generating        followed by incrementing Q by 1, said incrementing Q followed by        iterating from an index I=0 to I=C−1 in steps of 1, wherein        iteration I includes setting E=E₁ followed by executing the        program code recursively at a next level of recursion for the        node E;    -   else for each element in node E: incrementing P by 1, next        storing in A_(P) either U or the element pointing to U, and        lastly if the program code at all of said levels of recursion        has not been fully executed then resuming execution of said        program code at the most previous level of recursion at which        the program code was partially but not fully executed else        exiting the algorithm.

The present invention provides a method, computer program product, andassociated algorithm for sorting S sequences of binary bits in ascendingor descending order of a value associated with each sequence and in atime period denoted as a sorting execution time, said S sequences beingstored in a memory device of the computer system prior to said sorting,S being at least 2, each sequence of the S sequences comprising Kcontiguous fields denoted left to right as F₁, F₂, . . . , F_(K) withcorresponding field widths of W₁, W₂, . . . , W_(K), said sortingcomprising the steps of:

designating S memory areas of the memory device as A₁, A₂, . . . ,A_(S);

setting an output index P=0 and a field index Q=0;

providing a node E having S elements stored therein, said S elementsconsisting of the S sequences or S pointers respectively pointing to theS sequences; and

counter-controlled looping through program code, said looping includingiteratively executing said program code within nested loops, saidexecuting said program code including determining a truth or falsity ofan assertion that the elements in node E collectively include or pointto no more than one unique sequence U of the S sequences, and if saidassertion is determined to be false:

-   -   then generating C child nodes from node E, each child node        including all elements in node E having a unique value of field        F_(Q+1), said child nodes denoted as E₀, E₁, . . . , E_(C−1)        having associated field F_(Q+1) values of V₀, V₁, . . . ,        V_(C−1), said child nodes E₀, E₁, . . . , E_(C−1) being        sequenced such that V₀<V₁< . . . <V_(C−1); said generating        followed by incrementing Q by 1, said incrementing Q followed by        iterating from an index I=0 to I=C−1 in steps of 1, wherein        iteration I includes setting E=E₁ followed by returning to said        counter-controlled looping;    -   else for each element in node E: incrementing P by 1, next        storing in A_(P) either U or the element pointing to U, and        lastly if all iterations of said outermost loop have not been        executed then returning to said counter-controlled looping else        exiting from said algorithm.

The present invention advantageously provides a sort algorithm having anexecution speed of order less than N_(W) log N_(W).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a path through a linked execution structure, inaccordance with embodiments of the present invention.

FIG. 2 depicts paths through a linked execution structure for sortingintegers, in accordance with embodiments of the present invention.

FIG. 3 depicts FIG. 2 with the non-existent nodes deleted, in accordancewith embodiments of the present invention.

FIG. 4 depicts paths through a linked execution structure for sortingstrings with each path terminated at a leaf node, in accordance withembodiments of the present invention.

FIG. 5 is a flow chart for linear sorting under recursive execution, inaccordance with embodiments of the present invention.

FIG. 6 is a flow chart for linear sorting under counter-controlledlooping, in accordance with embodiments of the present invention.

FIGS. 7A-7D comprise source code for linear sorting of integers underrecursive execution, in accordance with embodiments of the presentinvention.

FIGS. 8A-8D comprise source code for linear sorting of strings underrecursive execution, in accordance with embodiments of the presentinvention.

FIG. 9 illustrates a computer system for sorting sequences of bits, inaccordance with embodiments of the present invention.

FIG. 10 is a graph depicting the number of moves used in sortingintegers for a values range of 0-9,999,999, using Quicksort and alsousing the linear sort of the present invention.

FIG. 11 is a graph depicting the number of compares used in sortingintegers for a values range of 0-9,999,999, using Quicksort and alsousing the linear sort of the present invention.

FIG. 12 is a graph depicting the number of moves used in sortingintegers for a values range of 0-9,999, using Quicksort and also usingthe linear sort of the present invention.

FIG. 13 is a graph depicting the number of compares used in sortingintegers for a values range of 0-9,999, using Quicksort and also usingthe linear sort of the present invention.

FIG. 14 is a graph depicting sort time used in sorting integers for avalues range of 0-9,999,999, using Quicksort and also using the linearsort of the present invention.

FIG. 15 is a graph depicting sort time used in sorting integers for avalues range of 0-9,999, using Quicksort and also using the linear sortof the present invention.

FIG. 16 is a graph depicting memory usage for sorting fixed-length bitsequences representing integers, using Quicksort and also using thelinear sort of the present invention.

FIG. 17 is a graph depicting sort time using Quicksort for sortingstrings, in accordance with embodiments of the present invention.

FIG. 18 is a graph depicting sort time using a linear sort for sortingstrings, in accordance with embodiments of the present invention.

FIGS. 19-24 is a graph depicting sort time used in sorting integers,using Quicksort and also using the linear sort of the present invention,wherein the sort time is depicted as a function of mask width andmaximum value that can be sorted.

DETAILED DESCRIPTION OF THE INVENTION

The detailed description is presented infra in three sections. The firstsection, in conjunction with FIG. 1, comprises an Introduction to thepresent invention, including assumptions, terminology, features, etc. ofthe present invention. The second section, in conjunction with FIGS. 2-9comprises a Sort Algorithm detailed description in accordance with thepresent invention. The third section, in conjunction with FIGS. 10-24,relates to Timing Tests, including a description and analysis ofexecution timing test data for the sort algorithm of the presentinvention in comparison with Quicksort.

Introduction

FIG. 1 depicts a path through linked execution structure, in accordancewith embodiments of the present invention. The linked executionstructure of FIG. 1 is specific to 12-bit words divided into 4contiguous fields of 3 bits per field. For example, the example word100011110110 shown in FIG. 1 is divided into the following 4 fields(from left to right): 100, 011, 110, 110. Each field has 3 bits andtherefore has a “width” of 3 bits. The sort algorithm of the presentinvention will utilize a logical mask whose significant bits (formasking purposes) encompass W bits. Masking a sequence of bits isdefined herein as extracting (or pointing to) a subset of the bits ofthe sequence. Thus, the mask may include a contiguous group of ones(i.e., 11 . . . 1) and the remaining bits of the mask are each 0; thesignificant bits of the mask consist of the contiguous group of ones,and the width W of the mask is defined as the number of the significantbits in the mask. Thus, W is referred to as a “mask width”, and the maskwidth W determines the division into contiguous fields of each word tobe sorted. Generally, if the word to be sorted has N bits and if themask width is W, then each word to be sorted is divided into L fields(or “levels”) such that L=N/W if N is an integral multiple of W, underthe assumption that the mask width W is constant. If N is not anintegral multiple of W, then the mask width cannot be constant. Forexample if N=12 and W=5, then the words to be sorted may be dividedinto, inter alia, 3 fields, wherein going from left to right the threefields have 5 bits, 5 bits, and 2 bits. In this example, L may becalculated via L=ceiling (N/W), wherein ceiling(x) is defined as thesmallest integer greater than or equal to x. Thus, the scope of presentinvention includes an embodiment in which W is a constant width withrespect to the contiguous fields of each word to be sorted.Alternatively, the scope of present invention also includes anembodiment in which W is a variable width with respect to the contiguousfields of each word to be sorted. Each word to be sorted may becharacterized by the same mask and associated mask width W, regardlessof whether W is constant or variable with respect to the contiguousfields.

Although the scope of the present invention permits a variable maskwidth W as in the preceding example, the example of FIG. 1 as well asthe examples of FIGS. 2-4 discussed infra use a constant mask width forsimplicity. For the example of FIG. 1, N=12, W=3, and L=4. It should benoted that the maximum numerical value that the N bits could have is2^(N)−1. Thus, the maximum value that a 12-bit word could have is 4095.

In FIG. 1, the linked execution structure has a root, levels, and nodes.Assuming a constant mask of width W, the root in FIG. 1 is representedas a generic field of W bits having the form xxx where x is 0 or 1.Thus, the width W of the mask used for sorting is the number of bits (3)in the root. The generic nodes corresponding to the root encompass allpossible values derived from the root. Hence the generic nodes shown inFIG. 1 are 000, 001, 010, 011, 011, 100, 101, 110, and 111. The numberof such generic nodes is 2^(W), or 8 if W==3 as in FIG. 1. There are Llevels (or “depths”) such that each field of a word corresponds to alevel of the linked execution structure. In FIG. 1, the 4 levels (i.e.,L=4) are denoted as Level 1, Level 2, Level 3, and Level 4.

Consider the example word 100011110110 shown in FIG. 1. Below the rootare 8 generic nodes of Level 1, called “child nodes” of the root. Thefirst field of the example word is 100 corresponding to the 100 node inLevel 1. Below the 100 node of Level 1 are the 8 generic nodes of Level2, namely the child nodes of the 100 node of Level 1. The second fieldof the example word is 011 corresponding to the 011 node in Level 2.Below the 011 node of Level 2 are its 8 child nodes in Level 3. Thethird field of the example word is 110 corresponding to the 110 node inLevel 3. Below the 110 node of Level 3 are its 8 child nodes in Level 4.The fourth field of the example word is 110 corresponding to the 110node in Level 4. Thus, the path through the linked execution structurefor the example word 100011110110 consists of the 100 node of level 1,the 011 child node of Level 2, the 110 child node of Level 3, and the110 child node of Level 4.

Although not shown in FIG. 1, each node of the linked executionstructure at level i potentially has the 2^(W) child nodes below it atlevel i+1. For example the 000 node at Level 1 has 8 child nodes belowit, and each such child nodes has 8 child nodes, etc. Thus the maximumnumber of nodes of the linked execution structure is2^(W)+2^(2W)+2^(3W)+ . . . +2^(LW), or (2^((L+1)W) ^(—)2^(W))/(2^(W)−1). In FIG. 1, the total number of nodes is 4680 for W=3and L=4. Since it is not practical to show all nodes of the linkedexecution structure, FIG. 1 shows only those nodes and their childrenwhich illustrate the path of the example word.

The actual nodes of a linked execution structure relative to a group ofwords to be sorted comprise actual nodes and non-existent nodes. Thepaths of the words to be sorted define the actual nodes, and theremaining nodes define the non-existent nodes. Thus in FIG. 1, theactual nodes include 100 node of level 1, the 011 child node of Level 2,the 110 child node of Level 3, and the 110 child node of Level 4. Anyother word having a path through the linked execution structure of FIG.1 defines additional actual nodes.

Another concept of importance is a “leaf node” of the linked executionstructure, which is an actual node that is also a terminal node of apath through the linked execution structure. A leaf node has nochildren. In FIG. 1, 110 node in Level 4 is a leaf node. In the contextof the sort algorithm of the present invention, it is also possible to aleaf node at a level other than the deepest Level L. Multiple numbers tobe sorted may give rise to a given node having more than one child(i.e., the paths of different numbers to be sorted may intersect in oneor more nodes). If a given node of the linked execution structure holdsmore than one unique word to be sorted, then the algorithm must processthe child nodes of the given node. If, however, the given node of thelinked execution structure holds no more than one unique word to besorted, then the given node is a leaf node and the sort algorithmterminates the path at the given node without need to consider the child(if any) of the given node. In this situation, the given node isconsidered to be a leaf node and is considered to effectively have nochildren. Thus, it is possible for a leaf node to exist at a level L₁wherein L₁<L. The concept of such leaf nodes will be illustrated by theexamples depicted in FIGS. 2-4, discussed infra.

The sort algorithm of the present invention has an execution time thatis proportional to N*Z, wherein Z is a positive real number such that1≦Z≦L. As stated supra, N is defined as the number of bits in each wordto be sorted, assuming that N is a constant and characterizes each wordto be sorted, wherein said assumption holds for the case of an integersort, a floating point sort, or a string sort such that the stringlength is constant. Z is a function of the distribution of leaf nodes inthe linked execution structure. The best case of Z=1 occurs if all leafnodes are at level 1. The worst case of Z=L occurs if all leaf nodesoccur at Level L. Thus, the execution time for the worst case isproportional to N*L, and is thus linear in N with L being a constantthat is controlled by a choice of mask width W. Therefore, the sortalgorithm of the present invention is designated herein as a “linearsort”. The term “linear sort” is used herein to refer to the sortingalgorithm of the present invention.

If the words to be sorted are strings characterized by a variable stringlength, then the execution time is proportional to Σ_(j)W_(j)N_(j),where N_(j) is a string length in bits or bytes (assuming that thenumber of bits per byte is a constant), wherein W_(j) is a weightingfactor that is proportional to the number of strings to be sorted havinga string length N_(j). The summation Σ_(j) is from j=1 to j=J such thatJ is the number of unique string lengths in the strings to be sorted.For example consider 60 strings to be sorted such that 30 strings have 3bytes each, 18 strings have 4 bytes each, and 12 strings have 5 byteseach. For this example, J=3, N₁=3 bytes, W₁∝30, N₂=4 bytes, W₂∝18, N₃=5bytes, W₃∝12 bytes, wherein the symbol “∝” stands for “proportional to”.Thus, the sort execution time is a linear combination of the stringlengths N_(j) (expressed in bits or bytes) of the variable-lengthstrings to be sorted. Accordingly, the sort algorithm of the presentinvention is properly designated herein as a “linear sort” for the caseof sorting variable-length strings.

In light of the preceding discussion, the sort algorithm of the presentinvention is designated herein as having a sorting execution time forsorting words (or sequences of bits), wherein said sorting executiontime is a linear function of the word length (or sequence length) of thewords (or sequences) to be sorted. The word length (or sequence length)may be a constant length expressed as a number of bits or bytes (e.g.,for integer sorts, floating point sorts, or string sorts such that thestring length is constant). Thus for the constant word length (orsequence length) case, an assertion herein and in the claims that thesorting execution time function is a linear function of the word length(or sequence length) of the words (or sequences) to be sorted means thatthe sorting execution time is linearly proportional to the constant wordlength (or sequence length).

Alternatively, the word length (or sequence length) may be a variablelength expressed as numbers of bits or bytes (e.g., for string sortssuch that the string length is variable). Thus for the constant wordlength (or sequence length) case, an assertion herein and in the claimsthat the sorting execution time function is a linear function of theword length (or sequence length) of the words (or sequences) to besorted means that the sorting execution time is proportional to a linearcombination of the unique non-zero values of string length (i.e.,N_(j)≠0) which characterize the strings to be sorted.

Note that the sorting execution time of the present invention is also alinear (or less than linear) function of S wherein S is the number ofsequences to be sorted, as will be discussed infra.

Also note that an analysis of the efficiency of the sorting algorithm ofthe present invention may be expressed in terms of an “algorithmiccomplexity” instead of in terms of a sorting execution time, inasmuch asthe efficiency can be analyzed in terms of parameters which the sortingexecution time depends on such as number of moves, number of compares,etc. This will be illustrated infra in conjunction with FIGS. 10-13.

As stated supra, L=N/W (if W is constant) and the upper-limiting valueV_(UPPER) that may potentially be sorted is 2^(N)−1. Consequently,L=(log₂V_(UPPER)+1)/W. Interestingly, L is thus dependent upon both Wand V_(UPPER) and does not depend on the number of values to be sorted,which additionally reduces the sort execution time. Inspection of thesort algorithm shows that a larger mask width W indicates a lessefficient use of memory but provides a faster sort except at the veryhighest values of W (see FIGS. 19-24 and description thereof). Since thesort execution time depends on W through the dependence of L or Z on W,one can increase the sort execution speed by adjusting W upward inrecognition of the fact that a practical upper limit to W may bedictated by memory storage constraints, as will be discussed infra.

The sort algorithm of the present invention assumes that: 1) for any twoadjacent bits in the value to be sorted, the bit to the left representsa larger magnitude effect on the value than the bit to the right; or 2)for any two adjacent bits in the value to be sorted, the bit to theright represents a larger magnitude effect on the value than the bit tothe left. The preceding assumptions permit the sort algorithm of thepresent invention to be generally applicable to integer sorts and stringsorts. The sort algorithm is also applicable to floating point sorts inwhich the floating point representation conforms to the commonly usedformat having a sign bit denoting the sign of the floating point number,an exponent field (wherein positive and negative exponents may bedifferentiated by addition of a bias for negative exponents as will beillustrated infra), and a mantissa field, ordered contiguously from leftto right in each word to be sorted. The sort algorithm is alsoapplicable to other data types such as: other floating pointrepresentations consistent with 1) and 2) above; string storage suchthat leftmost bytes represent the length of the string; little endianstorage; etc.

The sort algorithm of the present invention includes the followingcharacteristics: 1) the sort execution time varies linearly with N asdiscussed supra; 2) the sort execution time varies linearly (or lessthan linearly) with S as discussed supra; 3) the values to be sorted arenot compared with one another as to their relative values or magnitudes;4) the sort execution speed is essentially independent of the dataordering characteristics (with respect to data value or magnitude) inthe array of data to be sorted; 5) the sort efficiency (i.e., withrespect to execution speed) varies with mask width and the sortefficiency can be optimized through an appropriate choice of mask width;6) for a given mask width, sort efficiency improves as the data densityincreases, wherein the data density is measured by S/(V_(MAX)−V_(MIN)),wherein S denotes the number of values to be sorted, and wherein V_(MAX)and V_(MIN) are, respectively, the maximum and minimum values within thedata to be sorted, so that the sort execution time may vary less thatlinearly with S (i.e., the sort execution time may vary as S^(Y) suchthat Y<1); and 7) although the linked execution structure of FIG. 1underlies the methodology of the sort algorithm, the linked executionstructure is not stored in memory during execution of the sort (i.e.,only small portions of the linked execution structure are stored inmemory at any point during execution of the sort).

The linked execution structure of the present invention includes nodeswhich are linked together in a manner that dictates a sequential orderof execution of program code with respect to the nodes. Thus, the linkedexecution structure of the present invention may be viewed a programcode execution space, and the nodes of the linked execution structuremay be viewed as points in the program code execution space. As will beseen in the examples of FIGS. 2-4 and the flow charts of FIGS. 5-6,described infra, the sequential order of execution of the program codewith respect to the nodes is a function of an ordering of maskingresults derived from a masking of the fields of the words (i.e.,sequences of bits) to be sorted.

The Sort Algorithm

FIG. 2 depicts paths through a linked execution structure for sortingintegers, in accordance with embodiments of the present invention. FIG.2 illustrates a sorting method, using a 2-bit mask, for the eightintegers (i.e., S=8) initially sequenced in decimal as 12, 47, 44, 37,03, 14, 31, and 44. The binary equivalents of the words to be sorted areshown. Each word to be sorted has 6 bits identified from right to leftas bit positions 0, 1, 2, 3, 4, and 5. For this example: S=8, N=6, W=2,and L=3. The root is represented as a generic field of W=2 bits havingthe form xx where x is 0 or 1. The generic nodes corresponding to theroot are 00, 01, 10, and 11. The number of such generic nodes is 2^(W),or 4 for W=2 as in FIG. 2. There are 3 levels such that each field of aword to be sorted corresponds to a level of the linked executionstructure. In FIG. 2, the 3 levels (i.e., L=3) are denoted as Level 1,Level 2, and Level 3. A mask of 110000 is used for Level 1, a mask of001100 is used for Level 2, and a mask of 000011 is used for Level 3.

The Key indicates that a count of the number of values in each node isindicated with a left and right parenthesis ( ), with the exception ofthe root which indicates the form xx of the root. For example, the 00node of level one has three values having the 00 bits in bit positions 4and 5, namely the values 12 (001100), 03 (000011), and 14 (001110). TheKey also differentiates between actual nodes and non-existent nodes. Forexample, the actual 01 node in Level 1 is a leaf node containing thevalue 31, so that the nodes in Levels 2 and 3 that are linked to theleaf node 01 in Level 1 are non-existent nodes which are present in FIG.2 but could have been omitted from FIG. 2. Note that non-existent nodesnot linked to any path are omitted entirely from FIG. 2. For example,the non-existent 11 node in Level 1 has been omitted, since none of thewords to be sorted has 11 in bit positions 4 and 5. FIG. 3 depicts FIG.2 with all non-existent nodes deleted.

The integer sort algorithm, which has been coded in the C-programminglanguage as shown in FIG. 7, is applied to the example of FIG. 2 asfollows. An output array A(1), A(2), . . . , A(S) has been reserved tohold the outputted sorted values. For simplicity of illustration, thediscussion infra describes the sort process as distributing the valuesto be sorted in the various nodes. However, the scope of the presentinvention includes the alternative of placing pointers to values to besorted (e.g., in the form of linked lists), instead of the valuesthemselves, in the various nodes. Similarly, the output array A(1),A(2), . . . , A(S) may hold the sorted values or pointers to the sortedvalues.

The mask at each level is applied to a node in the previous level,wherein the root may be viewed as a root level which precedes Level 1,and wherein the root or root level may be viewed as holding the S valuesto be sorted. In FIG. 2 and viewing the root as holding all eight valuesto be sorted, the Level 1 mask of 110000 is applied to all eight valuesto be sorted to distribute the values in the 4 nodes (00, 01, 10, 11) inLevel 1 (i.e., based on the bit positions 4 and 5 in the words to besorted). The generic nodes 00, 01, 10, 11 are ordered in ascending value(i.e., 0, 1, 2, 3) from left to right at each of Levels 1, 2 and 3,which is necessary for having the sorted values automatically appearoutputted sequentially in ascending order of value. It is also necessaryto have the 11 bits in the mask shifted from left to right as theprocessing moves down in level from Level 1 to Level 2 to Level 3, whichis why the 11 bits are in bit positions 4-5 in Level 1, in bit positions2-3 in Level 2, and in bit positions 0-1 in Level 3. Applying the mask(denoted as “MASK”) to a word (“WORD”) means performing the logicaloperation MASK AND WORD to isolate all words having bits correspondingto “11” in MASK. As shown for Level 1, the 00 node has 3 values (12, 03,14), the 01 node has 1 value (31), the 10 node has 4 values (47, 44, 37,44), and the 11 node has zero values as indicated by the absence of the11 node at Level 1 in FIG. 2. Note that the 10 node in Level 1 hasduplicate values of 44. Next, the actual nodes 00, 01, and 10 in Level 1are processed from left to right.

Processing the 00 node of Level 1 comprises distributing the values 12,03, and 14 from the 00 node of Level 1 into its child nodes 00, 01, 10,11 in Level 2, based on applying the Level 2 mask of 001100 to each ofthe values 12, 03, and 14. Note that the order in which the values 12,03, and 14 are masked is arbitrary. However, it is important to trackthe left-to-right ordering of the generic 00, 01, 10, and 11 nodes asexplained supra. FIG. 2 shows that the 00 node of Level 2 (as linked tothe 00 node of Level 1) is a leaf node, since the 00 node of Level 2 hasonly 1 value, namely 03. Thus, the value 03 is the first sorted valueand is placed in the output array element A(1). Accordingly, the 00, 01,10, and 11 nodes of Level 3 (which are linked to the 00 node of Level 2which is linked to the 00 node of Level 1) are non-existent nodes. FIG.2 also shows that the 11 node of level 2 (as linked to the 00 node ofLevel 1) has the two values of 12 and 14. Therefore, the values 12 and14 in the 11 node of level 2 (as linked to the 00 node of Level 1) areto be next distributed into its child nodes 00, 01, 10, 11 of Level 3,applying the Level 3 mask 000011 to the values 12 and 14. As a result,the values 12 and 14 are distributed into the leaf nodes 00 and 10,respectively, in Level 3. Processing in the order 00, 01, 10, 11 fromleft to right, the value 12 is outputted to A(2) and the value 14 isoutputted to A(3).

FIG. 2 shows that the 01 node of Level 1 is a leaf node, since 31 is theonly value contained in the 01 node of Level 1. Thus, the value of 31 isoutputted to A(4). Accordingly, all nodes in Level 2 and 3 which arelinked to the 01 node of Level 1 are non-existent nodes.

Processing the 10 node of Level 1 comprises distributing the four values47, 44, 37, and 44 from the 10 node of Level 1 into its child nodes 00,01, 10, 11 in Level 2, based on applying the Level 2 mask of 001100 toeach of the values 47, 44, 37, and 44. FIG. 2 shows that the 01 node ofLevel 2 (as linked to the 10 node of Level 1) is a leaf node, since the01 node of Level 2 has only 1 value, namely 37. Thus, the value 37 isplaced in the output array element A(5). Accordingly, the 00, 01, 10,and 11 nodes of Level 3 which are linked to the 01 node of Level 2 whichis linked to the 10 node of Level 1 are non-existent nodes. FIG. 2 alsoshows that the 11 node of level 2 (as linked to the 10 node of Level 1)has the three values of 47, 44, and 44. Therefore, the values 47, 44,and 44 in the 11 node of level 2 (as linked to the 10 node of Level 1)are to be next distributed into its child nodes 00, 01, 10, 11 of Level3 (from left to right), applying the Level 3 mask 000011 to the values47, 44, and 44. As a result, the duplicate values of 44 and 44 aredistributed into the leaf nodes 00 in Level 3, and the value of 47 isdistributed into the leaf node 11 in level 3. Processing in the order00, 01, 10, 11 from left to right, the value 44 is outputted to A(6),the duplicate value 44 is outputted to A(7), and the value 47 isoutputted to A(8). Thus, the output array now contains the sorted valuesin ascending order or pointers to the sorted values in ascending order,and the sorting has been completed.

While the preceding discussion of the example of FIG. 2 considered thewords to be sorted to be integers, each of the words to be sorted couldbe more generally interpreted as a contiguous sequence of binary bits.The sequence of bits could be interpreted as an integer as was done inthe discussion of FIG. 2 supra. The sequence of bits could alternativelybe interpreted as a character string, and an example of such a characterstring interpretation will be discussed infra in conjunction with FIG.4. Additionally, the sequence could have been interpreted as a floatingpoint number if the sequence had more bits (i.e., if N were large enoughto encompass a sign bit denoting the sign of the floating point number,an exponent field, and a mantissa field). Thus, the sorting algorithm isgenerally an algorithm for sorting sequences of bits whoseinterpretation conforms to the assumptions stated supra. It should benoted, however, that if the sequences are interpreted as numbers (i.e.,as integers or floating point numbers) then the word length (in bits) Nmust be constant. If the sequences are interpreted as character strings,however, then the word length N is not required to be constant and thecharacter strings to be sorted may have a variable length.

An important aspect of the preceding sort process is that no comparisonswere made between the values to be sorted, which has the consequence ofsaving an enormous amount of processing time that would otherwise havebeen expended had such comparisons been made. The sort algorithm of thepresent invention accomplishes the sorting in the absence of suchcomparisons by the masking process characterized by the shifting of the11 bits as the processing moves down in level from Level 1 to Level 2 toLevel 3, together with the left to right ordering of the processing ofthe generic 00, 01, 10, 11 nodes at each level. The fact that the outputarray A(1), A(2), . . . , A(8) contains sorted values in ascending orderis a consequence of the first assumption that for any two adjacent bitsin the value to be sorted, the bit to the left represents a largermagnitude effect on the value than the bit to the right. If thealternative assumption had been operative (i.e., for any two adjacentbits in the value to be sorted, the bit to the right represents a largermagnitude effect on the value than the bit to the left), then the outputarray A(1), A(2), . . . , A(8) would contain the same values as underthe first assumption; however the sorted values in A(1), A(2), . . . ,A(8) would be in descending order.

The preceding processes could be inverted and the sorted results wouldnot change except possibly the ascending/descending aspect of the sortedvalues in A(1), A(2), . . . , (8). Under the inversion, the generic bitswould processed from right to left in the ordered sequence: 00, 01, 10,11 (which is equivalent to processing the ordered sequence 11, 10, 01,00 from left to right). As a result, the output array A(1), A(2), . . ., A(8) would contain sorted values in descending order as a consequenceof the first assumption that for any two adjacent bits in the value tobe sorted, the bit to the left represents a larger magnitude effect onthe value than the bit to the right. However under the inversion and ifthe alternative assumption had been operative (i.e., for any twoadjacent bits in the value to be sorted, the bit to the right representsa larger magnitude effect on the value than the bit to the left), thenthe output array A(1), A(2), . . . , A(8) would contain the sortedvalues in ascending order.

The preceding process assumed that the mask width W is constant. Forexample, W=2 for the example of FIG. 2. However, the mask width could bevariable (i.e., as a function of level or depth). For example consider asort of 16 bit words having mask widths of 3, 5, 4, 4 at levels 1, 2, 3,4, respectively. That is, the mask at levels 1, 2, 3, and 4 may be,inter alia, 1110000000000000, 0001111100000000, 000000001110000, and0000000000001111, respectively. Generally, for N-bit words to be sortedand L levels of depth, the mask widths W₁, W₂, . . . , W_(L)corresponding to levels 1, 2, . . . , L, respectively, must satisfy:W₁+W₂, + . . . +W_(L)≦N. It is always possible have masks such thatW₁+W₂, + . . . +W_(L)=N. However, an improvement in efficiency may beachieved for the special case in which all numbers to be sorted have 0in one or more contiguous leftmost bits, as will be illustrated infra.In said special case, said leftmost bits having 0 in all words to besorted would not be masked and consequently W₁+W₂, + . . . +W_(L)<N.

There are several reasons for having a variable mask width. A firstreason for having a variable mask width W is that it may not belogically possible to have a constant mask width if L>1, such as for thecase of N being a prime number. For example, if N=13, then there doesnot exist an integer L of at least 2 such that N/L is an integer. Intheory, it is potentially possible to choose W=N even if N is a primenumber. However, memory constraints may render the choice of W=Nunrealistic as will be discussed next.

A second reason for having a variable mask width W, even if it logicallypossible for W to be constant with L>1, is that having a variable W mayreduce the sort execution time inasmuch as the sort execution time is afunction of W as stated supra. As W is increased, the number of levelsmay decrease and the number of nodes to be processed may likewisedecrease, resulting in a reduction of processing time. However, the caseof sufficiently large W may be characterized by a smallest sortexecution time, but may also be characterized by prohibitive memorystorage requirements and may be impracticable (see infra FIG. 16 anddiscussion thereof). Thus in practice, it is likely that W can beincreased up to a maximum value above which memory constraints becomecontrolling. Thus the case of L>1 is highly likely, and two or more maskwidths will exist corresponding to two or more levels. As will be seenfrom the analysis of timing test data discussed in conjunction withFIGS. 19-24 discussed infra, the sort efficiency with respect toexecution speed is a function not only of mask width but also of thedata density as measured by S/(V_(MAX)−V_(MIN)). Moreover, the maskwidth and the data density do not independently impact the sortexecution speed. Instead the mask width and the data density are coupledin the manner in which they impact the sort execution speed. Therefore,it may be possible to fine tune the mask width as a function of level inaccordance with the characteristics (e.g., the data density) of the datato be sorted.

Another improvement in sort execution timing may result from finding thehighest or maximum value V_(MAX) to be sorted and then determine ifV_(MAX) is of such a magnitude that N can be effectively reduced. Forexample, if 8-bit words are to be sorted and V_(MAX) is determined tohave the value 00110101, then bits 7-8 of all words to be sorted have 00in the leftmost bits 6-7. Therefore, bits 7-8 do not have to beprocessed in the sorting procedure. To accomplish this, a mask could beemployed in a three-level sorting scheme having N=8, L=3, W₁=2, W₂=2 andW₃=2. The masks for this sorting scheme are 00110000 for level 1,00001100 for level 2, and 00000011 for level 3. Although N=8 technicallyprevails, the actual sort time will be reflective of N=6 rather thanN=8, because the masks prevent bits 6-7 from being processed.

Similarly, one could find a lowest or minimum value value V_(MIN) to besorted and then determine if V_(MIN) is of such a magnitude that N canbe effectively reduced. For example, if 8-bit words are to be sorted andV_(MIN) is determined to have the value 10110100, then bits 0-1 of allwords to be sorted have 00 in the rightmost bits 0-1. Therefore, bits0-1 do not have to be processed in the sorting procedure. To accomplishthis, a variable width mask could be employed in a three-level sortingscheme having N=8, L=3, W₁=2 W₁=2 and W₃=2. The masks for this sortingscheme are 11000000 for level 1, 00110000 for level 2, and 00001100 forlevel 3. Although N=8 technically in this scheme, the actual sort timewill be reflective of N=6 rather than N=8, because the masks preventbits 0-1 from being processed at all.

Of course, it may be possible to utilize both V_(MAX) and V_(MIN) in thesorting to reduce the effective value of N. For example, if 8-bit wordsare to be sorted and V_(MAX) is determined to have the value 00110100and V_(MIN) is determined to have the value 00000100, then bits 7-8 ofall words to be sorted have 00 in the leftmost bits 6-7 and bits 0-1 ofall words to be sorted have 00 in the rightmost bits 0-1. Therefore,bits 7-8 and 0-1 do not have to be processed in the sorting procedure.To accomplish this, a constant width mask could be employed in atwo-level sorting scheme having N=8, L=2, and W=2. The masks for thissorting scheme are 00110000 for level 1 and 00001100 for level 2.Although N=8 technically in this scheme, the actual sort time will bereflective of N=4 rather than N=8, because the masks prevent bits 6-7and 0-1 from being processed at all.

The integer sorting algorithm described supra in terms of the example ofFIG. 2 applies generally to integers. If the integers to be sorted areall non-negative, or are all negative, then the output array A(1), A(2),. . . , will store the sorted values (or pointers thereto) as previouslydescribed. However, if the values to be sorted are in a standard signedinteger format with the negative integers being represented as a two'scomplement of the corresponding positive integer, and if the integers tobe sorted include both negative and non-negative values, then outputarray A(1), A(2), . . . stores the negative sorted integers to the rightof the non-negative sorted integers. For example the sorted results inthe array A(1), A(2), . . . may appear as: 0, 2, 5, 8, 9, −6, −4, −2,and the algorithm could test for this possibility and reorder the sortedresults as: −6, −4, −2, 0, 2, 5, 8, 9.

The sorting algorithm described supra will correctly sort a set offloating point numbers in which the floating point representationconforms to the commonly used format having a sign bit, an exponentfield, and a mantissa field ordered contiguously from left to right ineach word to be sorted. The standard IEEE 754 format represents asingle-precision real number in the following 32-bit floating pointformat: Sign Bit (1 bit) Exponent Field (8 bits) Mantissa Field (23bits)

IEEE 754 requires the exponent field to have a +127 (i.e., 01111111)bias for positive exponents and no bias for negative exponents. Theexponent field bits satisfy the previously stated assumption that forany two adjacent bits in the value to be sorted, the bit to the leftrepresents a larger magnitude effect on the value than the bit to theright, as may be seen in the following table for the exponents of −2,−1, 0, +1, and +2. Exponent Value Exponent Field Bits −2 01111101 −101111110 0 01111111 1 10000000 2 10000001The number of bits in the exponent and mantissa fields in the aboveexample is merely illustrative. For example, the IEEE 754 representationof a double-precision floating point number has 64 bits (a sign bit, an11-bit exponent, and a 52-bit mantissa) subject to an exponent bias of+1023. Generally, the exponent and mantissa fields may each have anyfinite number of bits compatible with the computer/processor hardwarebeing used and consistent with the degree of precision desired. Althoughthe sign bit is conventionally 1 bit, the sort algorithm of the presentinvention will work correctly even if more than one bit is used todescribe the sign. It is assumed herein that the position of the decimalpoint is in a fixed position with respect to the bits of the mantissafield and the magnitude of the word is modulated by the exponent valuein the exponent field, relative to the fixed position of the decimalpoint. As illustrated supra, the exponent value may be positive ornegative which has the effect of shifting the decimal point to the leftor to the right, respectively.

Due to the manner in which the sign bit and exponent field affect thevalue of the floating-point word, a mask may used to define field thatinclude any contiguous sequence of bits. For example, the mask mayinclude the sign bit and a portion of the exponent field, or a portionof the exponent field and a portion of the mantissa field, etc. In the32-bit example supra, for example, the sorting configuration could have4 levels with a constant mask width of 8 bits: N=32, L=4, and W=8. Themask for level 1 is 111111110₂₄, wherein 0₂₄ represents 24 consecutivezeroes. The mask for level 2 is 00000000111111110₁₆, wherein 0₁₆represents 16 consecutive zeroes. The mask for level 3 is0₁₆1111111100000000. The mask for level 2 is 0₂₄11111111. Thus the maskfor level 1 includes the sign bit and the 7 leftmost bits of theexponent field, the mask at level 2 includes the rightmost bit of theexponent field and the 7 leftmost bits of the mantissa field, an themask for levels 3 and 4 each include 8 bits of the mantissa field.

If the floating point numbers to be sorted include a mixture of positiveand negative values, then the sorted array of values will have thenegative sorted values to the right of the positive sorted values in thesame hierarchical arrangement as occurs for sorting a mixture ofpositive and negative integers described supra.

FIG. 4 depicts paths through a linked execution structure for sortingstrings with each path terminated at a leaf node, in accordance withembodiments of the present invention. In FIG. 4, thirteen strings of 3bytes each are sorted. The 13 strings to be sorted are: 512, 123, 589,014, 512, 043, 173, 179, 577, 152, 256, 167, and 561. Each stringcomprises 3 characters selected from the following list of characters:0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each character consists of a byte,namely 8 bits. Although in the example of FIG. 4 a byte consists of 8bits, a byte may generally consist of any specified number of bits. Thenumber of potential children (i.e., child nodes) at each node is 2^(b)where b is the number of bits per byte. Thus in FIG. 4, each nodepotentially has 256 (i.e., 28) children. The sequence 014, 043, 123, . .. at the bottom of FIG. 4 denoted the strings in their sorted order.

In FIG. 4, the string length is constant, namely 3 characters or 24bits. Generally, however, the string length may be variable. Thecharacter string defines a number of levels of the linked executionstructure that is equal to the string length as measured in bytes. Thereis a one-to-one correspondence between byte number and level number. Forexample, counting left to right, the first byte corresponds to level 1,the second byte corresponds to level 2, etc. Thus, if the string lengthis variable then the maximum number of levels L of the linked executionstructure is equal to the length of the longest string to be sorted, andthe processing of any string to be sorted having a length less than themaximum level L will reach a leaf node at a level less than L.

The mask width is a constant that includes one byte, and the boundarybetween masks of successive levels coincide with byte boundaries.Although the sorting algorithm described in conjunction with the integerexample of FIG. 2 could be used to sort the character strings of FIG. 4,the sorting algorithm to sort strings could be simplified to takeadvantage of the fact that mask boundaries coincide with byteboundaries. Rather than using an explicit masking strategy, eachindividual byte may be mapped into a linked list at the byte'srespective level within the linked execution structure. Under thisscheme, when the processing of a string reaches a node corresponding tothe rightmost byte of the string, the string has reached a leaf node andcan then be outputted into the sorted list of strings. For example, aprogramming language with uses length/value pairs internally for stringstorage can compare the level reached with the string's length (inbytes) to determine when that the string has reached a leaf node. Thepreceding scheme is an implicit masking scheme in which the mask widthis equal to the number of bits in a character byte. Alternatively, thealgorithm could use an explicit masking scheme in which any desiredmasking configuration could be used (e.g., a mask could encompass bitsof two or more bytes). Thus, a masking strategy is always being used,either explicitly or implicitly.

In FIG. 4, the sorting of the thirteen strings 3-byte strings arecharacterized by S=13, N=24 (i.e. 3 bytes×8 bits/byte), W=8 (i.e., 1byte), and L=3. Shown in each node is a mask associated with the node,and the strings whose path passes through the node. The mask in eachnode is represented as a sequence of bytes and each byte might may beone of the following tluee unique symbols: X, x, and h where hrepresents one of the characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Theposition within the mask of the symbol X is indicative of the location(and associated level) of child nodes next processed. The X is used tomask various strings, as will be described infra, by setting X equal tothe mask character; thus if X is being used to isolate strings having“5” in the masked position of the strings then X=“5” will characterizethe mask. The symbol “h” and its position in the mask indicates that thestrings in the node each have the character represented by “h” in theassociated position. The position within the mask of the symbol “x”indicates the location (and associated level) of the mask representativeof other child nodes (e.g., “grandchildren”) to be subsequentlyprocessed.

The strings shown in each node in FIG. 4 each have the form H: s(1),s(2), . . . , wherein H represents a character of the string in the byteposition occupied by X, and wherein s(1), s(2), are strings having thecharacter represented by H in the byte position occupied by X. Forexample, in the node whose mask is 0Xx, the string denoted by 1:014 has“0” in byte position 1 and “1” in byte position 2, and the stringdenoted by 4:043 has “0” in byte position 1 and “4” in byte position 2.As another example, in the node whose mask is 17X, the string denoted by3:173 has “1” in byte position 1, “7” in byte position 2, and “3” inbyte position 3, whereas the string denoted by 9:179 has “1” in byteposition 1, “7” in byte position 2, and “9” in byte position 3.

The method of sorting the strings of FIG. 4 follows substantially thesame procedure as was described supra for sorting the integers of FIG.2. The string sort algorithm, which has been coded in the C-programminglanguage as shown in FIG. 8, is applied to the example of FIG. 4 asfollows. Similar to FIG. 2, an output array A(1), A(2), . . . , A(S) hasbeen reserved to hold the outputted sorted values. For simplicity ofillustration, the discussion infra describes the sort process asdistributing the values to be sorted in the various nodes. However, thescope of the present invention includes the alternative of placingpointers to values to be sorted (e.g., in the form of linked lists),instead of the values themselves, in the various nodes. Similarly, theoutput array A(1), A(2), . . . , A(S) may hold the sorted values orpointers to the sorted values.

First, the root node mask of Xxx is applied to all thirteen strings tobe sorted to distribute the strings in the 10 nodes 0Xx, 1Xx, . . . ,9Xx, resulting of the extraction and storage of the strings to be sortedand their identification with the first byte of 0, 1, 2, 3, 4, 5, 6, 7,8, or 9. Applying the mask a string may be accomplished by ANDing themask with the string to isolate the strings having a byte correspondingto the byte position of X in the mask to identify the child nodes. Asanother approach, the character bytes of a string could be pointed to orextracted from the string by use of a string array subscript, whereinthe string array subscript serves as the mask by providing thefunctionality of the mask. Masking a sequence of bits is defined hereinas extracting (or pointing to) a subset of the bits of the sequence.Thus, masking with X=0 isolates the strings 014 and 043 which definechild node 0Xx, masking with X=1 isolates the strings 123, 173, 179,152, 167 which defines the child node 1Xx, etc. Processing the Xxx rootnode comprises distributing the thirteen strings into the child nodes0Xx, 1Xx, etc. The child nodes 0Xx, 1Xx, etc. at Level 1 are nextprocessed on the order 0Xx, 1Xx, etc. since 0<1< . . . in charactervalue. Note that the characters are generally processed in the order 0,1, 2, . . . , 9 since 0<1<2< . . . in character value.

For the 0Xx node at level 1, the 0Xx mask is applied to the strings 014and 043 to define the next child nodes 01X and 04X, respectively, atLevel 2. The 01X and 04X nodes are processed in the sequential order of01X and 04X since 0 is less than 4 in character value. Note that thecharacters are always processed in the order 0, 1, 2, . . . , 9. The 01Xnode at Level 2 is processed, and since the 01X node contains only onestring, the 01X node is a leaf node and the string 014 is outputted toA(1). The 04X node at Level 2 is next processed and, since the 04X nodecontains only one string, the 04X node is a leaf node and the string 043is outputted to A(2).

For the 1Xx node at level 1, the 1Xx mask is applied to the strings 123,152, 167, (173, 179) to define the next child nodes 12X, 15X, 16X, and17X, respectively, at Level 2. The 12X, 15X, 16X, and 17X nodes areprocessed in the order 12X, 15X, 16X, and 17X, since the characters arealways processed in the order 0, 1, 2, . . . , 9 as explained supra. The12X node at Level 2 is processed, and since the 12X node contains onlyone string, the 12X node is a leaf node and the string 123 is outputtedto A(3). The 15X node at Level 2 is next processed and, since the 15Xnode contains only one string, the 15X node is a leaf node and thestring 152 is outputted to A(4). The 16X node at Level 2 is nextprocessed and, since the 16X node contains only one string, the 16X nodeis a leaf node and the string 167 is outputted to A(5). The 17X node atLevel 2 is next processed such that the 17X mask is applied to thestrings 173 and 179 to define the next child nodes 173 and 179 at Level3, which are processed in the order of 173 and 179 since 3 is less than9 in character value. The 173 node at Level 3 is next processed and,since the 173 node contains only one string, the 173 node is a leaf nodeand the string 173 is outputted to A(6). The 179 node at Level 3 is nextprocessed and, since the 179 node contains only one string, the 179 nodeis a leaf node and the string 179 is outputted to A(7).

For the 2Xx node at level 1, since the 2Xx node contains only onestring, the 2Xx node is a leaf node and the string 256 is outputted toA(8).

For the 5Xx node at level 1, the 5Xx mask is applied to the strings(512, 512), 561, 577, and 589 to define the next child nodes 51X, 56X,57X, and 58X, respectively, at Level 2. The 51X, 56X, 57X, and 58X nodesare processed in the order 51X, 56X, 57X, and 58X, since the charactersare always processed in the order 0, 1, 2, . . . , 9 as explained supra.The 512X node at Level 2 is processed; since the node 51X does notinclude more than one unique string (i.e., 512 appears twice asduplicate strings), the 51X node at Level 2 is a leaf node and theduplicate strings 512 and 512 are respectively outputted to A(9) and(10). The 56X node at Level 2 is next processed and, since the 56X nodecontains only one string, the 56X node is a leaf node and the string 561is outputted to A(11). The 57X node at Level 2 is next processed and,since the 57X node contains only one string, the 57X node is a leaf nodeand the string 577 is outputted to A(12). The 58X node at Level 2 isnext processed and, since the 58X node contains only one string, the 58Xnode is a leaf node and the string 589 is outputted to A(13). Thus, theoutput array now contains the sorted strings in ascending order of valueor pointers to the sorted values in ascending order of value, and thesorting has been completed.

Similar to the integer sort of FIG. 2, sorting the strings isessentially sorting the binary bits comprised by the strings subject toeach character or byte of the string defining a unit of mask. Thus, thesorting algorithm is generally an algorithm for sorting sequences ofbits whose interpretation conforms to the assumptions stated supra. Nocomparisons were made between the values of the strings to be sorted,which has the consequence of saving an enormous amount of processingtime that would otherwise have been expended had such comparisons beenmade. The output array A(1), A(2), . . . , A(13) contains sorted stringsin ascending order of value as a consequence of the first assumptionthat for any two adjacent bits (or bytes) in the string to be sorted,the bit (or byte) to the left represents a larger magnitude effect onthe value than the bit (or byte) to the right. If the alternativeassumption had been operative (i.e., for any two adjacent bits (orbytes) in the string to be sorted, the bit (or byte) to the rightrepresents a larger magnitude effect on the value than the bit (or byte)to the left), then the output array A(1), A(2), . . . , A(8) wouldcontain the same strings as under the first assumption; however thesorted values in A(1), A(2), . . . , A(8) would be in descending orderof value.

Similar to the integer sort of FIG. 2, the preceding processes could beinverted and the sorted results would not change except possibly theascending/descending aspect of the sorted strings in A(1), A(2), . . . ,(13). Under the inversion, the bytes 0, 1, 2, . . . , 8, 9 wouldprocessed from right to left in the ordered sequence: 0, 1, 2, . . . ,8, 9 (which is equivalent to processing the ordered sequence 9, 8, . . ., 2, 1, 0 from left to right). As a result, the output array A(1), A(2),A(8) would contain sorted strings in descending order of value is aconsequence of the first assumption that for any two adjacent bits (orbytes) in the string to be sorted, the bit (or byte) to the leftrepresents a larger magnitude effect on the value than the bit (or byte)to the right. However under the inversion and if the alternativeassumption had been operative (i.e., for any two adjacent bits (orbytes) in the value to be sorted, the bit (or byte) to the rightrepresents a larger magnitude effect on the value than the bit (or byte)to the left), then the output array A(1), A(2), . . . , A(8) wouldcontain the sorted strings in ascending order of value.

As seen from the examples of FIGS. 2-4, the linked execution structureof the present invention includes nodes which are linked together in amanner that dictates a sequential order of execution of program codewith respect to the nodes. Thus, the linked execution structure of thepresent invention may be viewed a program code execution space, and thenodes of the linked execution structure may be viewed as points in theprogram code execution space. Moreover, the sequential order ofexecution of the program code with respect to the nodes is a function ofan ordering of masking results derived from a masking of the fields ofthe words to be sorted.

FIG. 5 is a flow chart for linear sorting under recursive execution, inaccordance with embodiments of the present invention. The flow chart ofFIG. 5 depicts the processes described supra in conjunction with FIGS. 2and 4, and generally applies to sorting S sequences of binary bitsirrespective of whether the sequences are interpreted as integers,floats, or strings. Steps 10-12 constitute initialization, and steps13-20 are incorporated within a SORT module, routine, function, etc.which calls itself recursively in step 18 each time a new node isprocessed.

In step 10 of the initialization, the S sequences are stored in memory,S output areas A₁, A₂, . . . , A_(S) are set aside for storing thesorted sequences. S may be set to a minimum value such as, inter alia,2, 3, etc. The upper limit to S is a function of memory usagerequirements (e.g., see FIG. 16 and accompanying description) inconjunction with available memory in the computer system beingutilized). The output areas A₁, A₂, . . . , A_(S) correspond to theoutput areas A(1), A(2), . . . , A(S) described supra in conjunctionwith FIGS. 2 and 4. In addition an output index P and a field index Qare each initialized to zero. The output index P indexes the outputarray A₁, A₂, . . . , A_(S). The field index Q indexes field of asequence to be sorted, the field corresponding to the bits of thesequences that are masked and also corresponds to the levels of thelinked execution structure.

In step 11 of the initialization, the root node E₀ is initialized tocontain S elements associated with the S sequences. An element of asequence is the sequence itself or a pointer to the sequence inasmuch asthe nodes may contain sequences or pointers to sequences (e.g, linkedlists) as explained supra.

In step 12 of the initialization, a current node E is set equal to theroot node E₀. The current node E is the node that is currently beingprocessed. Initially, the current node E is the root node E₀ that isfirst processed.

SORT begins at step 13, which determines whether more than one uniqueelement is in the current node E being processed, is determining whetherE is a leaf node. No more than one unique element is in E if E contains1 or a plurality of identical elements, in which case E is a leaf node.If step 13 determines that there is no more than one unique element inE, then E is a leaf node and steps 14 and 15 are next executed. If step13 determines that there is more than one unique element in E, then nodeE is not a leaf node and step 16 is next executed.

Step 14 outputs the elements of E in the A array; i.e., for each elementin E, the output pointer P is incremented by 1 and the element is storedin A_(P).

Step 15 determines whether the sort is complete by determining whetherall nodes of the linked execution structure have been processed. Notingthat SORT calls itself recursively in step 18 each time a new node isprocessed and that the recursed call of SORT processes only the valuesassigned to the new node, it is clear that all nodes have been processedwhen a normal exit from the first node processed by SORT (i.e., the rootnode) has occurred. Thus step 15 effectuates a normal exit from SORT. Ifsaid normal exit from SORT is an exit from processing the root node bySORT, then the sorting has ended. Otherwise, step 20 effectuates areturn to execution of the previous copy of SORT that had beenrecursively executing. It should be noted that step 20 is notimplemented by explicit program code, but instead by the automaticbackward recursion to the previously executing version of SORT.

Step 16 is executed if E is not a leaf node. In step 16, the elements ofE are distributed into C child nodes: E₀, E₁, . . . E_(C−1), ascendinglysequenced for processing purposes. An example of this is in FIG. 4,wherein if E represents the root node Xxx then the elements of E (i.e.,the strings 014, 043, . . . , 577, 561) are distributed into the 4 childnodes (i.e., C=4) of 0Xx, 1Xx, 2Xx, and 5Xx. The child nodes areascendingly sequenced for processing, which means that the child nodesare processed in the sequence 0Xx, 1Xx, 2Xx, and 5Xx as explained suprain the discussion of FIG. 4.

Step 17 is next executed in which the field index Q (which is also thelevel index) is incremented by 1 to move the processing forward to thelevel containing the child nodes E₀, E₁, . . . E_(C−1). Step 15 alsoinitializes a child index 1 to 0. The child index points to the childnode E₁ (I=1, 2, . . . , L).

Steps 18-19 define a loop through the child nodes E₁, E₂, . . . E_(C).Step 18 sets the node E to E₁ and executes the SORT routine recursivelyfor node E. Thus the child node E₁ of the linked execution structure isa recursive instance of a point in the program code (i.e., SORT)execution space. When control returns (from the recursive call), thechild index I in incremented by 1, followed in step 19 by adetermination of whether the current child node E₁ being processed isthe last child to be processed (i.e., if I=C). If it is determined thatI≠C then execution return to the beginning of the loop at step 18 forexecution of the next child node. If it is determined that I=C then allchild nodes have been processed and step 20 is next executed. Step 20effectuates a return to execution of the previous copy of SORT that hadbeen recursively executing.

FIG. 6 is a flow chart for linear sorting under counter-controlledlooping, in accordance with embodiments of the present invention. FIG. 6effectuates the same sorting algorithm as FIG. 5, except that theprocedure of FIG. 5 executes the nodes recursively, while the procedureof FIG. 6 executes the nodes iteratively through counter-controlledlooping.

Step 31 provides initialization which may include substantially some orall of the processes executed in steps 10-12 if FIG. 5. Theinitializations in step 31 include storing the S sequences to be sorted,designating an output area for storing a sorted output array,initializing counters, etc. The number of sequences to be sorted (S) maybe set to a minimum value such as, inter alia, 2, 3, etc. The upperlimit to S is a function of memory usage requirements in conjunctionwith available memory in the computer system being utilized.

Step 32 manages traversal of the nodes of a linked execution structure,via counter-controlled looping. The order of traversal of the nodes aredetermined by the masking procedure described supra. Thecounter-controlled looping includes iterative execution of program codewithin nested loops. Step 32 controls the counters and the looping so asto process the nodes in the correct order; i.e., the order dictated bythe sorting algorithm depicted in FIG. 5 and illustrated in the examplesof 2 and 4. The counters track the nodes by tracking the paths throughthe linked execution structure, including tracking the level or depthwhere each node on each path is located. Each loop through the childrenof a level i node is an inner loop through nodes having a commonancestry at a level closer to the root. In FIG. 4, for example, an innerloop through the children 173 and 179 of node 17X at level 2 is innerwith respect to an outer loop through nodes 12X, 15X, 16X, and 16Xhaving the common ancestor of node 1Xx at level 1. Thus, the inner andouter loops of the preceding example form a subset of the nested loopsreferred to supra.

Since the paths are complex and each path is unique, the node countersand associated child node counters may be dynamically generated as theprocessing occurs. Note that the recursive approach of FIG. 5 alsoaccomplishes this tracking of nodes without the complexcounter-controlled coding required in FIG. 6, because the tracking inFIG. 5 is accomplished automatically by the compiler through compilationof the recursive coding. Thus from a programming effort point of view,the node traversal bookkeeping is performed in FIG. 5 by program codegenerated by the compiler's implementation of recursive calling, whereasthe node traversal bookkeeping is performed in FIG. 6 by program codeemploying counter-controlled looping explicitly written by a programmer.Using FIGS. 2, 4, and 5 as a guide, however, one of ordinary skill inthe art of computer programming can readily develop the required programcode (through counter-controlled looping) that processes the nodes inthe same order as depicted in FIGS. 2, 4, and 5 so as to accomplish thesorting according to the same fundamental method depicted in FIGS. 2, 4,and 5.

Step 33 determines whether all nodes have been processed, by determiningwhether all counters have attained their terminal values. Step 33 ofFIG. 6 corresponds to step 15 of FIG. 5. If all nodes have beenprocessed then the procedure ends. If all nodes have not been processedthen step 34 is next executed.

Step 34 establishes the next node to process, which is a function of thetraversal sequence through the linked execution structure as describedsupra, and associated bookkeeping using counters, of step 32.

Step 35 determines whether the node being processed is empty (i.e.,devoid of sequences to be sorted or pointers thereto). If the node isdetermined to be empty then an empty-node indication is set in step 36and the procedure loops back to step 32 where the node traversalmanagement will resume, taking into account the fact that the empty nodeindication was set. If the node is not determined to be empty then step37 is next executed. Note that steps 35 and 36 may be omitted if thecoding is structured to process only non-empty nodes.

Step 37 determines whether the node being processed is a leaf node(i.e., whether the node being processed has no more than one uniquesequence). Step 37 of FIG. 6 corresponds to step 13 of FIG. 5. If thenode is determined to be a leaf node then step 38 stores the sequences(or pointers thereto) in the node in the next available positions in thesorted output array, and a leaf-node indication is set in step 39followed by a return to step 32 where the node traversal management willresume, taking into account the fact that a leaf node indication wasset. If the node is not determined to be a leaf node then step 40 isnext executed.

Step 40 establishes the child nodes of the node being processed. Step 40of FIG. 6 corresponds to step 16 of FIG. 5

Step 41 sets a child nodes indication, followed by a return to step 32where the node traversal management will resume, taking into account thefact that a child nodes indication was set.

Note that the counter-controlled looping is embodied in steps 32-41through generating and managing the counters (step 32), establishing thenext node to process (step 34), and implementing program logic resultingfrom the decision blocks 33, 35, and 37.

Also note that although FIG. 6 expresses program logic natural tocounter-controlled looping through the program code, while FIG. 5expresses logic natural to recursive execution of the program code, thefundamental method of sorting of the present invention and theassociated key steps thereof are essentially the same in FIGS. 5 and 6.Thus, the logic depicted in FIG. 6 is merely illustrative, and thecounter-controlled looping embodiment may be implemented in any mannerthat would be apparent to an ordinary person in the art of computerprogramming who is familiar with the fundamental sorting algorithmdescribed herein. As an example, the counter-controlled loopingembodiment may be implemented in a manner that parallels the logic ofFIG. 5 with the exceptions of: 1) the counter-controlled looping throughthe program code replaces the recursive execution of the program code;and 2) counters associated with the counter-controlled looping need tobe programmatically tracked, updated, and tested.

FIGS. 7A, 7B, 7C, and 7D. (collectively “FIG. 7”) comprise source codefor linear sorting of integers under recursive execution and also fortesting the execution time of the linear sort in comparison withQuicksort, in accordance with embodiments of the present invention. Thesource code of FIG. 7 includes a main program (i.e., void main), afunction ‘build’ for randomly generating a starting array of integers tobe sorted), a function ‘linear sort’ for performing the linear sortalgorithm according to the present invention, and a function ‘quicksort’for performing the Quicksort algorithm. The ‘linear_sort’ function inFIG. 7B will be next related to the flow chart of FIG. 5.

Code block 51 in ‘linear_sort’ corresponds to steps 13-15 and 20 in FIG.5. Coding 52 within the code block 51 corresponds to step 20 of FIG. 5.

Code block 53 initializes the child array, and the count of the numberof children in the elements of the child array, to zero. Code block 53is not explicitly represented in FIG. 5, but is important forunderstanding the sort time data shown in FIGS. 19-24 described infra.

Code block 54 corresponds to step 16 in FIG. 5.

Coding 55 corresponds to I=I+1 in step 18 of FIG. 5, which shifts themask rightward and has the effect of moving to the next lower level onthe linked execution structure.

Coding block 56 corresponds to the loop of steps 18-19 in FIG. 5. Notethat linear_sort is recursively called in block 56 as is done instep 18of FIG. 5.

FIGS. 8A, 8B, 8C, and 8D (collectively “FIG. 8”) comprise source codefor linear sorting of strings under recursive execution and also fortesting the execution time of the linear sort, in comparison withQuicksort, in accordance with embodiments of the present invention. Thecoding in FIG. 8 is similar to the coding in FIG. 7. A distinction to benoted is that the coding block 60 in FIG. 8 is analogous to, butdifferent from, the coding block 54 in FIG. 7. In particular, block 60of FIG. 8 reflects that: a mask is not explicitly used but is implicitlysimulated by processing a string to be sorted one byte at a time; andthe string to be sorted may have a variable number of characters.

FIG. 9 illustrates a computer system 90 for sorting sequences of bits,in accordance with embodiments of the present invention, in accordancewith embodiments of the present invention. The computer system 90comprises a processor 91, an input device 92 coupled to the processor91, an output device 93 coupled to the processor 91, and memory devices94 and 95 each coupled to the processor 91. The input device 92 may be,inter alia, a keyboard, a mouse, etc. The output device 93 may be, interalia, a printer, a plotter, a computer screen, a magnetic tape, aremovable hard disk, a floppy disk, etc. The memory devices 94 and 95may be, inter alia, a hard disk, a dynamic random access memory (DRAM),a read-only memory (ROM), etc. The memory device 95 includes a computercode 97. The computer code 97 includes an algorithm for sortingsequences of bits in accordance with embodiments of the presentinvention. The processor 91 executes the computer code 97. The memorydevice 94 includes input data 96. The input data 96 includes inputrequired by the computer code 97. The output device 93 displays outputfrom the computer code 97. Either or both memory devices 94 and 95 (orone or more additional memory devices not shown in FIG. 9) may be usedas a computer usable medium having a computer readable program codeembodied therein, wherein the computer readable program code comprisesthe computer code 97.

While FIG. 9 shows the computer system 90 as a particular configurationof hardware and software, any configuration of hardware and software, aswould be known to a person of ordinary skill in the art, may be utilizedfor the purposes stated supra in conjunction with the particularcomputer system 90 of FIG. 9. For example, the memory devices 94 and 95may be portions of a single memory device rather than separate memorydevices.

Timing Tests

FIGS. 10-24, comprise timing tests for the sort algorithm of the presentinvention, including a comparison with Quicksort execution timing data.FIGS. 10-15 relate to the sorting of integers, FIG. 16 relates to memoryrequirement for storage of data, FIGS. 17-18 relate to the sorting ofstrings, and FIGS. 19-24 relate to sorting integers as a function ofmask width and maximum value that can be sorted. The integers to besorted in conjunction with FIGS. 10-15 and 19-24 were randomly generatedfrom a uniform distribution. The timing tests associated with FIGS.10-23 were performed using an Intel Pentium® III processor at 1133 MHz,and 512M RAM.

FIG. 10 is a graph depicting the number of moves versus number of valuessorted using a linear sort in contrast with Quicksort for sortingintegers for a values range of 0-9,999,999. The linear sort was inaccordance with embodiments of the present invention using the recursivesort of FIG. 5 as described supra. For counting the moves, a counter wasplaced in the linear algorithm and in Quicksort at each point where anumber is moved. Noting that 9,999,999 requires 24 bits to be stored,the linear sort was performed using mask widths W=2, 3, 4, 6, 8, 12, and14 with a corresponding number of levels L=12, 8, 6, 4, 3, 2, and 2,respectively. For cases in which 24 is not an integral multiple of W,the mask width was truncated in the rightmost field corresponding tolevel L (i.e., at the level furthest from the root). For example atW=14, the mask widths at levels 1 and 2 were 14 and 10, respectively,for a total of 24 bits. FIG. 10 shows that, with respect to moves for avalues range of 0-9,999,999, Quicksort is more efficient than the linearalgorithm for W=2, 3, and 4, whereas the linear algorithm is moreefficient than Quicksort for W=6, 8, 12, and 14.

FIG. 11 is a graph depicting the number of compares/moves versus numberof values sorted using a linear sort in contrast with Quicksort forsorting integers for a values range of 0-9,999,999. For the linear sort,the number of compares/moves is the same as the number of moves depictedin FIG. 10 inasmuch as the linear sort does not “compare” to effectuatesorting. For Quicksort, the number of compares/moves is a number ofcompares in addition to the number of moves depicted in FIG. 10. Thelinear sort was in accordance with embodiments of the present inventionusing the recursive sort of FIG. 5 as described supra. For counting thecompares, a counter was placed in the linear algorithm and in Quicksortat each point where a number is compared or moved. Noting that 9,999,999requires 24 bits to be stored, the linear sort was performed using maskwidths W=2, 3, 4, 6, 8, 12, and 14 with a corresponding number of levelsL=12, 8, 6, 4, 3, 2, and 2, respectively. For cases in which 24 is notan integral multiple of W, the mask width is truncated in the rightmostfield corresponding to level L. For example at W=14, the mask widths atlevels 1 and 2 were 14 and 10, respectively, for a total of 24 bits.FIG. 11 shows that, with respect to compares/moves for a values range of0-9,999,999, the linear algorithm is more efficient than Quicksort forall values of W tested.

FIG. 12 is a graph depicting the number of moves versus number of valuessorted using a linear sort in contrast with Quicksort for sortingintegers for a values range of 0-9,999. The linear sort was inaccordance with embodiments of the present invention using the recursivesort of FIG. 5 as described supra. For counting the moves, a counter wasplaced in the linear algorithm and in Quicksort at each point where anumber is moved. Noting that 9,999 requires 14 bits to be stored, thelinear sort was performed using mask widths W=2, 3, 4, 6, 8, 10, 12, 14with a corresponding number of levels L=7, 5, 4, 3, 2, 2, 2, and 1,respectively. For cases in which 14 is not an integral multiple of W,the mask width is truncated in the rightmost field corresponding tolevel L (i.e., in the cases of W=3, 4, 6, 8, 10, 12). FIG. 12 showsthat, with respect to moves for a values range of 0-9,999, Quicksort ismore efficient than the linear algorithm for W=2, 3, and 4, whereas thelinear algorithm is more efficient than Quicksort for W=6, 8, 10, 12,and 14.

FIG. 13 is a graph depicting the number of compares versus number ofvalues sorted using a linear sort in contrast with Quicksort for sortingintegers for a values range of 0-9,999. The linear sort was inaccordance with embodiments of the present invention using the recursivesort of FIG. 5 as described supra. For counting the compares, a counterwas placed in the linear algorithm and in Quicksort at each point wherea number is compared. Noting that 9,999 requires 14 bits to be stored,the linear sort was performed using mask widths W=2, 3, 4, 6, 8, 10, 12,14 with a corresponding number of levels L=7, 5, 4, 3, 2, 2, 2, and 1,respectively. For cases in which 14 is not an integral multiple of W,the mask width is truncated in the rightmost field corresponding tolevel L (i.e., in the cases of W=3, 4, 6, 8, 10, 12). FIG. 13 showsthat, with respect to compares for a values range of 0-9,999, the linearalgorithm is more efficient than Quicksort for all values of W tested.Of particular note is the difference in efficiency between the linearsort and Quicksort when the dataset contains a large number ofduplicates (which occurs when the range of numbers is 0-9,999 since thenumber of values sorted is much greater than 9,999). Because of theexponential growth of the number of comparisons required by theQuicksort, the test for sorting with multiple duplicates of values(range 0-9,999), the test had to be stopped at 6,000,000 numbers sorted.

FIG. 14 is a graph depicting the sort time in CPU cycles versus numberof values sorted using a linear sort in contrast with Quicksort forsorting integers for a values range of 0-9,999,999. The linear sort wasin accordance with embodiments of the present invention using therecursive sort of FIG. 5 as described supra. Noting that 9,999,999requires 24 bits to be stored, the linear sort was performed using maskwidths W=2, 3, 4, 6, 8, 10, 12, and 14 with a corresponding number oflevels L=12, 8, 6, 4, 3, 3, 2, and 2, respectively. For cases in which24 is not an integral multiple of W, the mask width was truncated in therightmost field corresponding to level L (i.e., at the level furthestfrom the root). For example at W=10, the mask widths at levels 1, 2, and3 were 10, 10, and 4, respectively, for a total of 24 bits. As anotherexample at W=14, the mask widths at levels 1 and 2 were 14 and 10,respectively, for a total of 24 bits. FIG. 14 shows that, with respectto sort time for a values range of 0-9,999,999, Quicksort is moreefficient than the linear algorithm for W=2, 3, and 4, whereas thelinear algorithm is more efficient than Quicksort for W=6, 8, 10, 12,and 14.

FIG. 15 is a graph depicting the sort time in CPU cycles versus numberof values sorted using a linear sort in contrast with Quicksort forsorting integers for a values range of 0-9,999. The linear sort was inaccordance with embodiments of the present invention using the recursivesort of FIG. 5 as described supra. Noting that 9,999 requires 14 bits tobe stored, the linear sort was performed using mask widths W=2, 3, 4, 6,8, 10, 12, and 14 with a corresponding number of levels L=7, 5, 4, 3, 2,2, 2, and 1, respectively. For cases in which 24 is not an integralmultiple of W, the mask width was truncated in the rightmost fieldcorresponding to level L (i.e., in the cases of W=3, 4, 6, 8, 10, 12.FIG. 15 shows that, with respect to sort time for a values range of0-9,999, the linear algorithm is more efficient than Quicksort for allvalues of W tested, which reflects the large number of compares for datahaving many duplicate values as discussed supra in conjunction with FIG.13.

FIG. 16 is a graph depicting memory usage using a linear sort incontrast with Quicksort for sorting 1,000,000 fixed-length sequences ofbits representing integers, in accordance with embodiments of thepresent invention using the recursive sort of FIG. 5 as described supra.Quicksort is an in-place sort and therefore uses less memory than doesthe linear sort. The linear sort uses memory according to the followinggeneral formula, noting that this formula focuses only on the mainmemory drivers of the algorithm:MEM=S*M _(V)+(M _(C)*2^(W−1) *L)wherein MEM is the number of bytes required by the linear sort, S is thenumber of sequences to be sorted, Mv is the size of the data structure(e.g., 12) required to hold each sequence being sorted, M_(C) is thesize of the data structure (e.g., 8) required to hold a child sequenceor pointer in the recursive linked execution structure, W is the widthof the mask (≧1), and L is the number of levels of recursion. For someembodiments, L=ceiling(M_(V)/W) as explained supra.

In FIG. 16, M_(V)=12 and M_(C)=8. The Quicksort curve in FIG. 16 isbased on Quicksort using 4 bytes of memory per value to be sorted. Thegraphs stops at a mask width of 19 because the amount of memory consumedwith the linear sort approaches unrealistic levels beyond that point.Thus, memory constraints serve as upper limit on the width of the maskthat can be used for the linear sort.

FIGS. 17 and 18 graphically depict the sort time in CPU cycles versusnumber of strings sorted for the linear sort and Quicksort,respectively. The linear sort was in accordance with embodiments of thepresent invention using the recursive sort of FIG. 5 as described supra.The tests were conducted with simple strings. A file of over 1,000,000strings was created by extracting text-only strings from such sources aspublic articles, the Bible, and various other sources. Each set of testswas run against strings ranging up to 20 characters in length (maxlen=20) and then again against strings ranging up to 30 characters inlength (max_len=30). A set of tests is defined as sorting a collectionof 10,000 strings and repeating the sort with increasing numbers ofstrings in increments of 10,000. No sorting test was performed on morethan 1,000,000 strings.

Quicksort is subject to chance regarding the value at the “pivot” pointsin the list of strings to be sorted. When unlucky, Quicksort is forcedinto much deeper levels of recursion (>200 levels). Unfortunately, thiscaused stack overflows and the tests abnormally terminated at 430,000strings sorted by Quicksort. By reordering the list of strings,Quicksort could be made to complete additional selections, but thenumber of tests completed were sufficient to demonstrate the comparisonof the linear sort versus the quicksort. FIGS. 17 and 18 shows that,with respect to sort time, the linear algorithm is more efficient thanQuicksort by a factor in a range of about 30 to 200 if the number ofstrings sorted is at least about 100,000.

Another distinction between the linear sort and Quicksort is that inQuicksort the string comparisons define extra loops, which adds amultiplier A, resulting in the Quicksort execution time having adependence of A*S*log S such that A is the average length of the string.The average length A of the string is accounted for in the linear sortalgorithm as the number of levels L.

FIGS. 17 and 18 demonstrate that the linear sort far outperformsQuicksort for both max_len=20 and max_len=30, and at all values of thenumber of strings sorted. A primary reason for the difference betweenthe linear sort and Quicksort is that Quicksort suffers from a “levelsof similarity” problem as the strings it is sorting become increasinglymore similar. For example, to differentiate between “barnacle” and“break”, the string compare in the linear sort examines only the first 2bytes. However, as Quicksort recurses and the strings becomeincreasingly more similar (as with “barnacle” and “barney”), increasingnumbers of bytes must be examined with each comparison. Combining thesuperlinear growth of comparisons in Quicksort with the increasing costsof each comparison produces an exponential growth effect for Quicksort.Evidence of the effect of increasingly more costly comparisons inQuicksort can be understood by noting that the number of compares andmoves made by the Quicksort are the same even though the maximum lengthof strings increases from 20 to 30. However, the number of clock cyclesrequired to perform the same number of moves and comparisons inQuicksort increases (see FIG. 17) as the maximum length of stringsincreases from 20 to 30, because the depth of the comparisons increases.FIG. 18 shows that the increase from 20 to 30 characters in the maximumlength of strings affects the number of clock cycles for the linearsort, because the complexity of the linear sort is based on the size ofthe data to be sorted. The lack of smoothness in the Quicksort curves ofFIG. 17 arises because of the sensitivity of Quicksort to the initialordering of the data to be sorted, as explained supra.

FIGS. 19-24 is a graph depicting sort time using a linear sort, incontrast with Quicksort, for sorting integers as a function of maskwidth and maximum value that can be sorted, in accordance withembodiments of the present invention. The values of S in FIGS. 19-24 aresignificantly smaller than the values of S used in FIGS. 10-15 and17-18. The linear sort was in accordance with embodiments of the presentinvention using the recursive sort of FIG. 5 as described supra. In eachof FIGS. 19-24, Time in units of CPU cycles is plotted versus MAX WIDTHand MOD_VAL, wherein MAX WIDTH (equivalent to W discussed supra) is thewidth of the mask, and wherein the integer values to be sorted wererandomly generated from a uniform distribution between 0 and MOD_VAL−1.Also in each of FIGS. 19-24, MAX WIDTH=13 is the rightmost arrayrepresenting Quicksort and has nothing to do with a mask width. LettingS denote the number of integer values sorted in each test, S=2000 inFIGS. 19-20, S=1000 in FIGS. 21-22, and S=100 in FIGS. 23-24. FIGS. 19and 20 represent the same tests and the scale of the Time directiondiffers in FIGS. 19 and 20. FIGS. 21 and 22 represent the same tests andthe scale of the Time direction differs in FIGS. 21 and 22. FIGS. 23 and24 represent the same tests and the scale of the Time direction differsin FIGS. 23 and 24. A difference between the tests of FIGS. 19-24 andthe tests of FIGS. 10-16 is that much fewer values are sorted in FIGS.19-24 than in FIGS. 10-16.

FIGS. 19-24 show a “saddle” shape effect in the three-dimensional Timeshape for the linear sort. The saddle shape is characterized by: 1) fora fixed MOD_VAL the Time is relatively high at low values of MASK WIDTHand at high values of MASK WIDTH but is relatively small at intermediatevalues of MASK WIDTH; and 2) for a fixed MASK WIDTH, the Time increasesas MOD_VAL increases.

Letting W denote MASK WIDTH, the effect of W on Time for a fixed MOD_VALis as follows. The Time is proportional to the product of the averagetime per node and the total number of nodes. The average time per nodeincludes additive terms corresponding to the various blocks in FIG. 7B,and block 53 is an especially dominant block with respect to computationtime. In particular, block 53 initializes memory in a time proportionalto the maximum number of child nodes (2^(W)) per parent node. Let Arepresent the time effects in the blocks of FIG. 7B which are additiveto the time (∝2^(W)) consumed by block 53. It is noted that 2^(W)increases monotonically and exponentially as W increases. However, thetotal number of nodes is proportional to N/W where N is the number ofbits in each word to be sorted. It is noted that 1/W decreasesmonotonically as W increases. Thus the behavior of Time as a function ofW depends on the competing effects of (2^(W)+A) and 1/W in theexpression (2^(W)+A)/W. This results in the saddle shape noted supra asW varies and MOD_VAL is held constant.

It is noted that the dispersion or standard deviation σ is inverse tothe data density as measured by S/(V_(MAX)−V_(MIN)), wherein S denotesthe number of values to be sorted, and V_(MAX) and V_(MIN) respectivelydenote the maximum and minimum values to be sorted. For FIGS. 19-24,V_(MIN)≧0 and V_(MAX)≦MOD_VAL−1. Thus, for a fixed data density of the Svalues, the Time is a saddle-shaped function of a width W of the mask.Although, FIGS. 19-24 pertain to the sorting of integers, the executiontime of the linear sorting algorithm of the present invention forsorting sequences of bits is essentially independent of whether thesequences of bits are interpreted as integers or floating point numbers,and the execution time is even more efficient for string sorts than forinteger sorts as explained supra. Therefore, generally for a fixed datadensity of S sequences of bits to be sorted, the sorting execution timeis a saddle-shaped function of a width W of the mask that is used in theimplementation of the sorting algorithm.

At a fixed mask width W and a fixed number of values S to be sorted,increasing MOD VAL increases the dispersion or standard deviation σ ofthe data to be sorted. Increasing σ increases the average number ofnodes which need to be processed in the sorting procedure. However, theTime increases as the average number of nodes needed to be processedincreases. This results in the increase in Time as MOD_VAL increaseswhile W is fixed. As to Quicksort, FIGS. 19-24 show that Time alsoincreases as MOD_VAL increases for Quicksort.

A corollary to the preceding analyses is that for a fixed W, thestandard deviation σ decreases (or the data density increases) as Sincreases, so that for a fixed W the sort execution time may vary lessthat linearly with S (i.e., the sort execution time may vary as S^(Y)such that Y<1).

FIGS. 19-24 show that for a given number S of values to be sorted, andfor a given value of MOD_VAL, there are one or mode values of W forwhich the linear sort Time is less than the Quicksort execution time. Apractical consequence of this result is that for a given set of data tobe sorted, said data being characterized by a dispersion or standarddeviation, one can choose a mask width that minimizes the Time and thereis one or more values of W for which the linear sort Time is less thanthe Quicksort execution time.

Although FIGS. 19-24 shows timing tests data for sorting integers, theability to choose a mask resulting in the linear sort of the presentinvention executing in less time than a sort using Quicksort alsoapplies to the sorting of floating point numbers since the linear sortalgorithm is essentially the same for sorting integers and sortingfloating point numbers. Additionally, the ability to choose a maskresulting in the linear sort executing in less time than a sort usingQuicksort also applies to the sorting of character strings inasmuch asFIGS. 14-15 and 17-18 demonstrate that the sorting speed advantage ofthe linear sort relative to Quicksort is greater for the sorting ofstrings than for the sorting of integers. It should be recalled that themask used for the sorting of character strings has a width equal to abyte representing a character of the string.

While embodiments of the present invention have been described hereinfor purposes of illustration, many modifications and changes will becomeapparent to those skilled in the art. Accordingly, the appended claimsare intended to encompass all such modifications and changes as fallwithin the true spirit and scope of this invention.

1. A computer program product, comprising: a computer usable medium having a computer readable program code embodied therein, said computer readable program code comprising an algorithm for sorting S input sequences of binary bits in ascending or descending order of a value associated with each sequence and in a time period denoted as a sorting execution time, said S sequences being stored in a memory device of a computer system prior to said sorting, S being at least 2, each sequence of the S sequences comprising K contiguous fields denoted left to right as F₁, F₂, . . . , F_(K) with corresponding field widths of W₁, W₂, . . . , W_(K), said algorithm adapted to perform said sorting by executing the steps of: designating S memory areas of the memory device as A₁, A₂, . . . , A_(S); setting an output index P=0 and a field index Q=0; providing a node E having S elements stored therein, said S elements consisting of the S sequences or S pointers respectively pointing to the S sequences; and executing program code, including determining a truth or falsity of an assertion that the elements in node E collectively include or point to no more than one unique sequence U of the S sequences, and if said assertion is determined to be false: then generating C child nodes from node E, each child node including all elements in node E having a unique value of field F_(Q+1), said child nodes denoted as E₀, E₁, . . . , E_(C−1) having associated field F_(Q+1) values of V₀, V₁, . . . , V_(C−1), said child nodes E₀, E₁, . . . , E_(C−1) being sequenced such that V₀<V< . . . <V_(C−1), said generating followed by incrementing Q by 1, said incrementing Q followed by iterating from an index I=0 to I=C−1 in steps of 1, wherein iteration I includes setting E=E₁ followed by executing the program code recursively at a next level of recursion for the node E; else for each element in node E: incrementing P by 1, next storing in A_(P) either U or the element pointing to U, and lastly if the program code at all of said levels of recursion has not been fully executed then resuming execution of said program code at the most previous level of recursion at which the program code was partially but not fully executed else exiting the algorithm.
 2. The computer program product of claim 1, wherein said algorithm is not adapted to execute comparing a value of a first sequence of the S sequences with a value of a second sequence of the S sequences.
 3. The computer program product of claim 1, wherein the sorting execution time is a linear function of a sequence length comprised by each sequence of the S sequences.
 4. The computer program product of claim 1, wherein the sorting execution time is a linear or less than linear function of S.
 5. The computer program product of claim 1, wherein the sorting execution time is essentially independent of an extent to which the S sequences are ordered in the memory device, prior to said sorting, with respect to said associated values.
 6. The computer program product of claim 1, wherein the sorting execution time is a decreasing function of a data density of the S sequences.
 7. The computer program product of claim 1, wherein the S elements consist of the S pointers, wherein the node E having the S sequences prior to said sorting includes a linked list that comprises the S pointers, and wherein each child node having pointers therein includes a linked list that comprises said pointers therein.
 8. The computer program product of claim 1, wherein the S sequences each represent a variable-length character string, wherein each of the S character strings consists of the K contiguous fields, wherein K is a sequence-dependent variable subject to W₁=W₂= . . . =W_(K)=one byte consisting of a fixed number of binary bits for representing one character.
 9. The computer program product of claim 1, wherein the S sequences each represent a fixed-length character string, wherein each of the S character strings consists of the K contiguous fields, wherein K is a sequence-dependent variable subject to W₁=W₂= . . . =W_(K)=one byte consisting of a fixed number of binary bits for representing one character.
 10. The computer program product of claim 1, wherein the S sequences consist of S fixed-length words such that each of the S words has N binary bits, wherein N is at least
 2. 11. The computer program product of claim 10, wherein the S words each represent an integer.
 12. The computer program product of claim 10, wherein the algorithm is further adapted to execute determining a leftmost significant bit position of the S words collectively, and wherein the leftmost bit position of field F₁ is the leftmost significant bit position of the S words collectively.
 13. The computer program product of claim 10, wherein the S words each represent a floating point number having the following fields contiguously ordered from left to right: a sign field, an exponent field, and a mantissa field.
 14. The computer program product of claim 1, wherein generating the C child nodes from node E comprises performing (M AND X) or (X AND M) with a mask M for each sequence X in node E, wherein the mask M is keyed to the field F_(Q+1), and wherein the bit positions of the mask M relating to the field F_(Q+1) each have a 1 bit, and wherein the remaining bit positions of the mask M each have a 0 bit.
 15. The computer program product of claim 1, wherein S is at least 1000, and wherein W₁, W₂, . . . , W_(K) is such that the sorting execution time is less than a Quicksort execution time for sorting the S sequences via execution of a Quicksort sorting algorithm by said processor.
 16. A computer program product, comprising: a computer usable medium having a computer readable program code embodied therein, said computer readable program code comprising an algorithm for sorting S input sequences of binary bits in ascending or descending order of a value associated with each sequence and in a time period denoted as a sorting execution time, said S sequences being stored in a memory device of a computer system prior to said sorting, S being at least 2, each sequence of the S sequences comprising K contiguous fields denoted left to right as F₁, F₂, . . . , F_(K) with corresponding field widths of W₁, W₂, . . . , W_(K), said algorithm adapted to perform said sorting by executing the steps of: designating S memory areas of the memory device as A₁, A₂, . . . , A_(S); setting an output index P=0 and a field index Q=0; providing a node E having S elements stored therein, said S elements consisting of the S sequences or S pointers respectively pointing to the S sequences; and counter-controlled looping through program code, said looping including iteratively executing said program code within nested loops, said executing said program code including determining a truth or falsity of an assertion that the elements in node E collectively include or point to no more than one unique sequence U of the S sequences, and if said assertion is determined to be false: then generating C child nodes from node E, each child node including all elements in node E having a unique value of field F_(Q+1), said child nodes denoted as E₀, E₁, . . . , E_(C−1) having associated field F_(Q+1) values of V₀, V₁, . . . , V_(C−1), said child nodes E₀, E₁, . . . , E_(C−1) being sequenced such that V₀<V₁< . . . <V_(C−1); said generating followed by incrementing Q by 1, said incrementing Q followed by iterating from an index I=0 to I=C−1 in steps of 1, wherein iteration I includes setting E=E₁ followed by returning to said counter-controlled looping; else for each element in node E: incrementing P by 1, next storing in A_(P) either U or the element pointing to U, and lastly if all iterations of said outermost loop have not been executed then returning to said counter-controlled looping else exiting from said algorithm.
 17. The computer program product of claim 16, wherein said algorithm is not adapted to execute comparing a value of a first sequence of the S sequences with a value of a second sequence of the S sequences.
 18. The computer program product of claim 16, wherein the sorting execution time is a linear function of a sequence length comprised by each sequence of the S sequences.
 19. The computer program product of claim 16, wherein the sorting execution time is a linear or less than linear function of S.
 20. The computer program product of claim 16, wherein the sorting execution time is essentially independent of an extent to which the S sequences are ordered in the memory device, prior to said sorting, with respect to said associated values.
 21. The computer program product of claim 16, wherein the sorting execution time is a decreasing function of a data density of the S sequences.
 22. The computer program product of claim 16, wherein the S sequences each represent a variable-length character string, wherein each of the S character strings consists of the K contiguous fields, wherein K is a sequence-dependent variable subject to W₁=W₂= . . . =W_(K)=one byte consisting of a fixed number of binary bits for representing one character.
 23. The computer program product of claim 16, wherein the S sequences each represent a fixed-length character string, wherein each of the S character strings consists of the K contiguous fields, wherein K is a sequence-dependent variable subject to W₁=W₂= . . . =W_(K)=one byte consisting of a fixed number of binary bits for representing one character.
 24. The computer program product of claim 16, wherein the S sequences consist of S fixed-length integers such that each of the S integers has N binary bits, wherein N is at least
 2. 25. The computer program product of claim 16, wherein the S sequences consist of S fixed-length floating point numbers, each of said floating point numbers having the following fields contiguously ordered from left to right: a sign field, an exponent field, and a mantissa field. 